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Codes from Goppa codes

Chunlei Liu

TL;DR

It is proved that these new codes approach the Gilbert-Varshamov bound and can be decoded within O(n^2(\logn)^a) operations in the symbol field, which is usually much small than the location field, where $n$ is the codeword length, and $a$ a constant determined by the polynomial factorization algorithm.

Abstract

On a Goppa code whose structure polynomial has coefficients in the symbol field, the Frobenius acts. Its fixed codewords form a subcode. Deleting the naturally occurred redundance, we obtain a new code. It is proved that these new codes approach the Gilbert-Varshamov bound. It is also proved that these codes can be decoded within $O(n^2(\logn)^a)$ operations in the symbol field, which is usually much small than the location field, where $n$ is the codeword length, and $a$ a constant determined by the polynomial factorization algorithm.

Codes from Goppa codes

TL;DR

It is proved that these new codes approach the Gilbert-Varshamov bound and can be decoded within O(n^2(\logn)^a) operations in the symbol field, which is usually much small than the location field, where is the codeword length, and a constant determined by the polynomial factorization algorithm.

Abstract

On a Goppa code whose structure polynomial has coefficients in the symbol field, the Frobenius acts. Its fixed codewords form a subcode. Deleting the naturally occurred redundance, we obtain a new code. It is proved that these new codes approach the Gilbert-Varshamov bound. It is also proved that these codes can be decoded within operations in the symbol field, which is usually much small than the location field, where is the codeword length, and a constant determined by the polynomial factorization algorithm.
Paper Structure (6 sections, 19 theorems, 66 equations)

This paper contains 6 sections, 19 theorems, 66 equations.

Table of Contents

  1. INTRODUCTION
  2. Construction
  3. Lower bounds on minimal distance and dimension
  4. The Gilbert-Varshamov bound
  5. Decoding
  6. Comparison with Goppa codes

Key Result

Lemma 2.3

The Goppa code $\Gamma(g,\delta,m,q)$ is stable under the Fröbenius.

Theorems & Definitions (33)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 3.1: distance bound
  • Lemma 3.2
  • Lemma 3.3: codeword equations
  • ...and 23 more